I would recommend working at FSI as an instructor if you are offered a job. To me, one of the best reasons to go to an academy type school is because you want to get a job there. At FSI, you should get a steady stream of motivated students and a lot of multi-time. You are also paid decently and get full benefits. So, although I was not always happy in my time there, compared to many FBO operations, it was a paradise. Half of your dual given will probably be multi-time if you work there. It depends on how many CFI students you do after you complete the 400 dual given requirement to do CFI initials.
If you do get hired I would not leave until you are close to the ATP minimums, but if you complete the 800 hour contract, you will have at least 1000 hours. I would then go to one of the FSI learning centers. The minimum to get typed at FSI is 1000 hours, so you would have that box checked. If you wait until you have your ATP minimums, then you can get your ATP as well. The important thing is the cross country hours. Instructing in the Seminole makes it doable to get the 500 cc requirement out of the way. You can count 100 hours in the sim toward the 1500 hr ATP minimum, so total time is not as important as the cc time.
The reason I suggest being an instructor first, is because it is very hard to get looked at in the corporate world with out your ATP. If you have an ATP and a type rating, you are much more marketable. The other thing I would recommend is be realistic when making your aircraft choices. I would recommend against going for the GV or the Global, just because it is a long shot. There was a guy that was hired to fly the GV out of the internship with very low hours, but that was an anomaly. Most insurance companies would choke.
So, my advice is try to be an instructor at FSI first, then do the internship. If that does not work out then by all means do the internship. I don't know how Snow is doing, but I averaged $20k per year instructing at FSI (this was after 9/11 when we went from over 100 instructor to 35, others were not so lucky). At the center, I usually worked at least 30 hours per week, sometimes considerably more, so I made in the mid 20's. I did some contract work as well, after I was there a while. I usually got $500 per day, although I would go lower at first, when I had no jet time.
If you do contract work, get your rate in writing (at least an e-mail), and make sure they understand it is for all days away from home. I did a 3 day contract in the Beechjet which consisted of one day to airline to the jet, one day of flying and a final day to airline home. Paycheck = $500, when I call they said they only pay for flying. We compromised and they gave me another $500. Then they want to know if I want a job! No thank you.
If you do decide to go this route, you will meet a lot of people. You will be decently compensated at the Centers and will learn more than I can tell you. You will also get to see what it is like being a sim instructor, which may be something you will consider if you lose your medical or want to be home more later in your career. If you do your job, one of the people you meet will hire you and you will get a job that is significantly better than going to the reqionals.